Finite Group Databasebeta

← #121505

Subgroups — #121505

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21986C211yesmaximal, characteristic
2#21365C211yesmaximal, characteristic
3#13955C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5#8033C211yesmaximal, characteristic
6C2×#1285C211yesmaximal, characteristic
7#7904C211yesmaximal, characteristic
8C2×#432C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×C4×M16C2211yescharacteristic
11#2285C2211yescharacteristic
12C2×#382C2211yescharacteristic
13C2×C4×C4⋊C4C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1949—22no—
16#1948—21no—
17#1947—21no—
18#1853—21no—
19#1853—21no—
20#1852—21no—
21#1852—21no—
22#1847—21no—
23#1836—21no—
24#1776—22no—
25C2×#339—22no—
26#1442—21no—
27#1442—21no—
28C4⋊#95—21no—
29C4⋊#95—21no—
30#1434—22no—
31#1295—22no—
32#1285C414yescomplemented
33#1279—22no—
34C22×M16—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D411yescharacteristic
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42D412yes—
43C22×C42C2311yescharacteristic
44#432—21no—
45#432—21no—
46#432—21no—
47#432—21no—
48#432C2×C412yes—
49#432C2×C412yes—
50#432Q812yes—
51#432Q812yes—
52C2×#107—22no—
53C2×#107—22no—
54C2×#106—22no—
55C2×#106—22no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8D412yes—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C4×M16—21no—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16Q812yes—
64C2×#97—21no—
65C2×#97—21no—
66C2×#97D412yes—
67C2×C4×C8—21no—
68#384D411yescharacteristic
69#384D411yescharacteristic
70#382—21no—
71#382—21no—
72#382—21no—
73#382—21no—
74#382—21no—
75#380—21no—
76#380—21no—
77C4×C4⋊C4—21no—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4C2×C412yes—
86C4×C4⋊C4Q812yes—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#339—44no—
100C23×C4—22no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—41no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117#117—42no—
118#117—42no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C22⋊C4—41no—
142C2×C42—22no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42—42no—
146C2×C42—42no—
147C2×C42—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42C4⋊C414yes—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q811yescharacteristic
156C2×C42C2×Q811yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42C2×Q811yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163#107—44no—
164#107—82no—
165#106—44no—
166#106—82no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8C4⋊C414yes—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175#97—22no—
176#97—22no—
177#97—22no—
178#97—22no—
179#97—41no—
180#97C4⋊C414yes—
181C4×C8—22no—
182C4×C8—41no—
183#95—22no—
184#95—22no—
185#95—44no—
186#95—44no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194#95—41no—
195#95—82no—
196#95—82no—
197C24#12011yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4C2×C4⋊C411yescharacteristic
240C22×C4C4×D411yescharacteristic
241C22×C4C4×D411yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4#12011yescharacteristic
244C22×C4#12111yescharacteristic
245C22×C4#12111yescharacteristic
246C22×C4#12111yescharacteristic
247C22×C4C22⋊Q811yescharacteristic
248C22×C4C22⋊Q811yescharacteristic
249C22×C4C22⋊Q811yescharacteristic
250C22×C4C22⋊Q811yescharacteristic
251C22×C4C22⋊Q811yescharacteristic
252C22×C4#12311yescharacteristic
253C22×C4#12711yescharacteristic
254C22×C4C4⋊Q811yescharacteristic
255C22×C4C4⋊Q811yescharacteristic
256C22×C4C4⋊Q811yescharacteristic
257M16—24no—
258M16—42no—
259M16—42no—
260M16—42no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—44no—
293C42—82no—
294C42—82no—
295C42—82no—
296C42—82no—
297C42C2×C4⋊C411yescharacteristic
298C42C2×C4⋊C411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×Q811yescharacteristic
301C42C22⋊Q811yescharacteristic
302C42C22⋊Q811yescharacteristic
303C42C22⋊Q811yescharacteristic
304C42C22⋊Q811yescharacteristic
305C42#12311yescharacteristic
306C42#12311yescharacteristic
307C42#12511yescharacteristic
308C42#12511yescharacteristic
309C42#12511yescharacteristic
310C42C4⋊Q811yescharacteristic
311C42C4⋊Q811yescharacteristic
312C42C4⋊Q811yescharacteristic
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23#38511yescharacteristic
318C23#39211yescharacteristic
319C23#39511yescharacteristic
320C23#45311yescharacteristic
321C23#45411yescharacteristic
322C23#45511yescharacteristic
323C23#45611yescharacteristic
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#38411yescharacteristic
388C2×C4#38511yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38911yescharacteristic
391C2×C4#38911yescharacteristic
392C2×C4#38911yescharacteristic
393C2×C4#39511yescharacteristic
394C2×C4#39511yescharacteristic
395C2×C4#39811yescharacteristic
396C2×C4#39811yescharacteristic
397C2×C4#39811yescharacteristic
398C2×C4#39911yescharacteristic
399C2×C4#45412yes—
400C2×C4#45512yes—
401C8—44no—
402C8—44no—
403C8—44no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22#144411yescharacteristic
417C22#185611yescharacteristic
418C22#185711yescharacteristic
419C22#200311yescharacteristic
420C22#200411yescharacteristic
421C22#200611yescharacteristic
422C22#200711yescharacteristic
423C22#201011yescharacteristic
424C22#201011yescharacteristic
425C22#206011yescharacteristic
426C22#206111yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—84no—
438C4—82no—
439C4—82no—
440C4—161no—
441C4—161no—
442C4—161no—
443C4—161no—
444C4#186012yes—
445C4#206112yes—
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2#1000511yescharacteristic
451C2#1000611yescharacteristic
452C2#1000911yescharacteristic
453C2#1001011yescharacteristic
454C2#1133511yescharacteristic
455C2#1133911yescharacteristic
456C2#1134211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
32. #1285C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#219862
#213652
#139552
#108512
#80332
C2×#12852
#79042